Below 105 K, the crystal structure of SrTiO\(_3\)  forms a tetragonal unit cell with the oxygen octahedra rotated with respect to the cubic cell.
This rotation counteracts the formation of ferroelectricity and is hence usually referred to as an anti-ferro-distortive (AFD) motion [1]}.
Therefore, the tetragonal geometry can be described as a \(\sqrt{2} \times \sqrt{2} \times 2\)  supercell of the primitive cubic perovskite ABO\(_3\)  unit cell with an additional AFD rotation, as depicted in Figs. 1(a) and 1(b).
This AFD in-plane rotation is accompanied by an elongation and a contraction of the \(c\)  and \(a\)  lattice vectors relative to the cubic structure [1]}.
To investigate the optimized geometry and total energy, we perform DFT calculations using the Quantum Espresso package [3]}.
The projector augmented wave method is employed to describe core level atomic orbitals and a plane-wave basis set with 70 Ry energy cut-off is used.
The Brillouin zone is sampled with \(6 \times 6 \times 4\)  \(\mathbf {k}\) -points.
In Tab. 1, we summarize the lattice parameter \(a\) , \(c/a\)  ratio and AFD rotation angle obtained with various DFT functionals.
Comparing with experimental observation [4]}, [5]}, [6]}, [7]}, local density approximation (LDA) [8]} and Perdew-Berke-Ernzerhof revised for solid (PBEsol) [9]} functionals provide a contracted lattice parameter \(a\)  and a higher \(a/c\)  ratio with an over-rotated AFD angle.
Even though the Perdew-Berke-Ernzerhof (PBE) [10]} functional and Heyd-Scuseria-Ernzerhof (HSE06) [11]}, [12]} hybrid functional describe elongated \(a\)  and \(c\)  lattices with over-rotated AFD angle, these lattice parameters are closer to experimental observations than the former two functionals.
